Circle Group - Group Structure

Group Structure

In this section we will forget about the topological structure of the circle group and look only at its structure as an abstract group.

The circle group T is a divisible group. Its torsion subgroup is given by the set of all nth roots of unity for all n, and is isomorphic to Q/Z. The structure theorem for divisible groups tells us that T is isomorphic to the direct sum of Q/Z with a number of copies of Q. The number of copies of Q must be c (the cardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum of c copies of Q is isomorphic to R, as R is a vector space of dimension c over Q. Thus

The isomorphism

can be proved in the same way, as C× is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of T.